Polarization quaternion DOA estimation based on vector MISC array

SHAO Shuai ,LIU Aijun,* ,YU Changjun ,and ZHAO Quanrui

1.School of Electronics and Information Engineering,Harbin Institute of Technology,Harbin 150001,China;2.Shanghai Radio Equipment Research Institute,Shanghai 201109,China

Abstract:This paper examines the direction of arrival (DOA) estimation for polarized signals impinging on a sparse vector sensor array which is based on the maximum interelement spacing constraint (MISC).The vector array effectively utilizes the polarization domain information of incident signals,and the quaternion model is adopted for signals polarization characteristic maintenance and computational burden reduction.The features of MISC arrays are crucial to the mutual coupling effects reduction and higher degrees of freedom (DOFs).The quaternion data model based on vector MISC arrays is established,which extends the scalar MISC array into the vector MISC array.Based on the model,a quaternion multiple signal classification (MUSIC)algorithm based on vector MISC arrays is proposed for DOA estimation.The algorithm combines the advantages of the quaternion model and the vector MISC array to enhance the DOA estimation performance.Analytical simulations are performed to certify the capability of the algorithm.

Keywords:direction of arrival (DOA),vector sensor array,vector maximum interelement spacing constraint (MISC) array,polarization quaternion model,quaternion multiple signal classification (Q-MUSIC).

1.Introduction

Array signal processing is a basic theory in the fields of radar,sonar,navigation,etc [1,2].With the increasing reliability of vector sensors,polarization is added to the direction of arrival (DOA) estimation as a basic information attribute. Therefore,many researchers proposed multi-component data processing algorithms.For polarization vector sensor arrays,multiple signal classification(MUSIC)-like algorithms were introduced in [3] and estimation of signal parameter via rotational invariance techniques (ESPRIT) were introduced in [4].Nehorai et al5] studied the Cramer-Rao bound (CRB) for the vector-sensor arrays.In [6],a multiscale disambiguation algorithm based on a multiscale sparse array of spatially spread electromagnetic vector-sensor (SS-EMVS) array was developed to attain high accuracy and unambiguous DOA estimations.In [7],a joint estimation of DOA and polarization of the signals was attained based on SSEMVS array,which can efficaciously decrease the complexity of the four-dimensional space joint estimation.Cao et al8] achieved the DOA estimation of correlated/coherent signals using a uniform rectangular array (URA)with electromagnetic vector-sensors.In [9],the proposed algorithm based on a sparse rectangular spatially spread crossed-dipole polarization sensitive array attained the two-dimensional (2D) DOA estimation which used the lowest components polarized vector sensor array and retained low mutual coupling.Song et al10] proposed a method which utilized the signals,inherent time-frequency peculiarity to obtain better performance of DOA estimation with a few snapshots for the underlying vectorsensor array in a noisy and coherent situation.In [11],Yang et al.proposed a hybrid L-shaped array of two sparse scalar arrays and a single triangular SS-EMVS for 2D DOA estimation,which obtained a better estimation performance.Ahmed et al12] proposed a fast quadrilinear decomposition algorithm for DOA and polarization estimation by an EMVS-URA,which kept rapid convergence and accurate parameter estimation.In [13],an estimation algorithm of 2D DOA and polarization parameter was proposed based on the three-parallel co-prime polarization sensitive array,which enhanced the estimation accuracy even in low signal-to-noise (SNR) ratio situation.

However,these methods presume that the complex valued data model represents incident signal frequency domain samples.The data covariance matrix is next described as second order statistical magnitude between all sensor components.In the recent few decades,algorithms based on quaternion were introduced [14] for array signal processing.A multidimensional complex signals hypercomplex version [15] was also introduced by Bülow et al16].In [17],Miron et al.proposed a quaternion data model algorithm based on vector-sensor array for estimation of DOA and polarization parameters,which decreased the data covariance model representation memory size leading to an efficient algorithm.In [18],for Lshaped array,the dimension reduction Q-MUSIC algorithm was proposed to decrease the complexity.In[19],a generalized rotation invariance based quaternion multiple signal classification algorithm was proposed for the estimation of polarization parameters and DOA,which not only reduced the computation complexity considerably,but also avoided the performance degradation caused by the failure in parameters pairing.The DOA could be estimated by biquaternion MUSIC in [20] to achieve better angle measurement performance.A quaternion-based ESPRIT-type algorithm called augmented quaternion ESPRIT was proposed for DOA estimation with a co-located crossed-dipole array in [21] to achieve better performance in low SNR scenarios.

The main advantages of sensor arrays are their spatial selectivity and their ability to decrease interference and enhance signal quality.Traditional uniform linear arrays(ULAs) are commonly used sensor arrays,in which the spacing between elements does not exceed half wavelength and is constant to avert spatial aliasing.However,the traditional ULA degrees of freedom (DOFs) are merely linear to the sensor number.For a ULA ofNsensors,the traditional subspace methods [22] may distinguish up toN−1 signals.For heightening the traditional ULA DOFs,additional sensors are needed,resulting in an extra complexity.Traditional ULAs also endure serious mutual coupling influence from array sensors.The sparse arrays,nonuniform linear arrays (NLAs),provide an interesting resolution for these difficulties.In sparse arrays,the mutual coupling influence can be decreased because of the interelement spacing increase [23].Latterly,the sparse arrays advancement,for instance nested arrays[24] and coprime arrays (CAs) [25],have attracted researchers to review this subject because sparse arrays can be methodically devised and their DOFs can be analytically offered from closed-form formulas.

The nested array can provideO(N2) DOFs with merelyNsensors,which is achieved via increasing interelement spacing and incorporating two or more ULAs [24].The two-level nested array is composed of two ULAs.However,the nested array DOFs become lower than that of the minimum redundancy array for a given sensors number.Yang et al26] introduced an improved nested array(INA) by enlarging the outer ULA interelement spacing and adding an additional sensor.The INA offer higher DOFs and retain all the two-level nested array advantages.The CA is another interesting sparse array.The CA[25] includes two ULAs in which the first ULA isN′-sensor array withinterelement spacing,while the second ULA isN′′-sensor array withinterelement spacing,where λ is signal wavelength andN′andN′′are coprime integers.For the nested array,the mutual coupling is further decreased in the CA.However,for the same number of physical sensors,the DOFs provided via a coprime array usually become smaller than that via a nested array.For further enlarging the DOFs,Pal et al27] proposed an extended CA via doubling the sensor number in an ULA.Its difference coarray may achieve consecutive lags between (−N′N′′−N′′+1) and(N′N′′+N′′−1) via using (N′+N′′−1) sensors.In [28],the CAs were derived from achieving two procedures.One procedure is constricting the subarray interelement spacing in the CA providing a compressed interelement spacing (CACIS). The CACIS normally enlarges the unique and consecutive lags number.To avoid a considerable number of overlaps of self-lags and cross-lags,in[28],the second procedure was proposed for displacing a subarray in the CA in order to offer a larger array aperture,a higher unique lags number,and a much larger minimum interelement spacing from the obtained CA with displaced subarrays.In [29],Zheng introduced an advanced array based on the maximum interelement spacing constraint (MISC) principle,which provided a higher DOFs and mutual coupling influence reduction.

At present,MISC arrays are only composed of scalar sensors and fail to display the polarization information of signals.The MISC array is composed of vector sensors,because it takes advantage of the array manifold,and takes into account the signal polarization information,which brings more benefits to DOA estimation.Based on the above analysis,this paper proposes a quaternion MUSIC based on vector MISC array.

This paper consists of the following sections.In Section 2,short quaternion descriptions are introduced and a polarized signal model is given.The MISC array concept is introduced and the characteristics of MISC array are discussed in Section 3.In Section 4,a description of the quaternion DOA estimation based on vector MISC array is given.In Section 5,the algorithm performances are evaluated by computer simulations.Finally,the conclusions are presented in Section 6.

2.Quaternion data model of polarized signals

2.1 Polarization model

Quaternions discovered by Hamilton in 1843 are four-dimensional (4D) hypercomplex numbers,which are complex numbers extensions into 4D space.A quaternion is defined via one real and three imaginaries,whose Cartesian form can be described as

where i2=j2=k2=ijk=−1,ij=k,ji=−k,ki=j,ik=−j,jk=i,kj=−i.

Many books researched quaternions and their characteristics.The hypercomplex numbers basics were introduced in [30],while a comprehensive quaternions review was introduced in [31].Several complex numbers properties may be extended into quaternions: the conjugateq∗=w−xi−yj−zk;pure quaternionq=xi+yj+zk;quaternion modulus∥q∥=quaternion inverseq−1=.H The quaternions set,noted ,forms a noncommutative normed algebra,meaning

And conjugation over H is an anti-involution

In [32],a Cayley-Dickson form quaternion was expressed as:q=q1+q2j,whereq1=w+xi andq2=y+zi .That is

The joint expression of spatial domain and polarization domain of signals is modeled and the signal receiving model of electromagnetic vector sensor array is established.This paper mainly studies fully polarized electromagnetic waves.

Assuming that there is a point signal source with elevation angleθand azimuth angleϕat a certain point in space.The ideal transmission medium is transmitted to the array along the direction of the Poynting vector,as shown inFig.1.

The unit vectors pointing in the positive direction ofx,y,andzaxis are denoted asex,ey,andezrespectively.The coordinate origin serves as the reference phase point of the electromagnetic vector sensor array,and the electromagnetic waves received are converted into baseband electrical signals.

As can be seen fromFig.1,the unit vector of the incident direction of the signal is the propagation vector of the signal

Fig.1 Characterization of fully polarized electromagnetic waves

where the superscript T represents the transpose.

The horizontal vectoreHand the vertical vectoreVconstitute a set of standard orthogonal basis perpendicular to the plane of the propagation direction:

For the electromagnetic wave at pointPin space,the complex analytical formula of the instantaneous electric field vector is

where

Equation (9) is the instantaneous projection of the complex analytical formula of the electric field vector in the horizontal and the vertical directions at timet.EHmandEVmare maximums in the horizontal and the vertical directions,respectively.The signal polarization information may be described via the instantaneous ratio of the electric field amplitude and phase in the vertical and the horizontal directions,and tanγ=η=φV−φH,is polarization assist angle,and η ∈[0,2π] is polarization phase difference.For a fully polarized wave,its endpoint polarization trajectory is an ellipse with a fixed long and short axial ratio and inclination.That is

The coordinates of the electric field vector in the plane rectangular coordinate system are

The coordinates of the magnetic field in the plane rectangular coordinate system are

The coordinates of a complete six-dimensional electric and magnetic field in a plane rectangular coordinate system are

where

AP(ϕ,θ,γ,η) is the polarization steering vector and β(ϕ,θ) shows the dependence of the polarization domain scan vector on the azimuth and elevation of the electromagnetic wave.

There is a uniform linearNelectromagnetic vector sensor array along the positive direction of they-axis.The electromagnetic vector sensor at the origin is used as the reference element.The electric dipoles of the electric field are orthogonal to each other,and the distance between the electromagnetic vector sensor array elements isd,andd<,which is used to ensure that there is no array blur,that is,the expected direction of the incident signal is only one value.Whenis taken as the value,the array element utilization rate is the highest and the effect is guaranteed.λis the wavelength of the incident desired signal.They-axis coordinateynof thenth electromagnetic vector sensor is (n−1)d,n=1,2,···,N,inFig.2.And makeM(M

Fig.2 Uniform linear array

The steering vector is

The data received by the entire array is

wherenn(t)=[nny(t)nnz(t)]T,(n=1,2,···,N) is the noise vector in directions ofyandz. ⊗ represents the Kronecker product.It is assumed that the noise is independent between each array element,the noise is independent between the vector components within each array element,and the signal and noise are relatively independent.Noise is an independent Gaussian white noise with a zero mean and aσ2variance.

2.2 Quaternions data model

Take theYandZdirection electric field components in(14) to obtain the polarization domain steering vector of themth signal received by this electromagnetic vector sensor array

The elevationθ=90°,and after simplification

Apply (4) here.The quaternion Cayley-Dickson representation synthesizes the polarization domain steering vector in the complex number domain into a polarization domain steering parameter in the quaternion domain,namely

After simplification,(21) becomes

According to (18),it can be known that themth signal received by the entire electromagnetic vector sensor array is

where

The quaternion noise is synthesized from the complex noise vector by Cayley-Dickson representation as

All signals received by the entire array can be expressed as

where

Makeam=As(ϕm)p(ϕm,γm,ηm),and write (26) as a matrix

Ais called the spatial-polarization joint steering vector matrix.In this way,a signal receiving data model of the electromagnetic vector sensor array in the quaternion domain is established.

3.MISC array

3.1 Difference coarray

Sensor positions of anN-sensor nonuniform linear array are given bynnumd,in whichnnumbelongs to an integer setS={nnum,num=1,2,···,N} andd=denotes the minimum distance between sensors.According to (24),themth signal array steering vector is

where

The data model is still (28),but the steering vector is(29).The covariance matrix ofX(t) may be given by

where E{·} is the statistical expectation operator,

is the source covariance matrix,the superscripts ◁ represents the quaternion conjugate transpose,diag(a) represents a diagonal matrix whose diagonal elements area,INdenotes theN×Nidentity matrix.

VectorizingRXXyields

where vec(·) represents the vectorization operator by stacking all columns of a matrix into a column vector one by one,B=,b(ϕm)=andwith 1mbeing a column zeros vector except a 1 at themth position.By comparing (28) with (33),the vectorzcan be regarded as the data received from the coherent source signal vectorpwith a single snapshot,and 2σ21 becomes a deterministic noise term.The different rows ofBbehave similarly to the manifold of a virtual array with expanded aperture,and the positions of its sensors are given by a set of differencesDin Definition 1.This virtual array is known as the original array difference coarray [33].Assuming thatU=[−Lu,Lu]is theDconsecutive segment,the corresponding measurements can be rearranged as

whereLuis the maximum of consecntive segment,Jrepresents a (2Lu+1)×N2selection matrix,B′is a(2Lu+1)×Mconsecutive virtual ULAUmanifold matrix,and 1' is a(2Lu+1)×1 zeros vector except a 1 at the(Lu+1)th position.The steering vector of the original matrix is shown in (24),and that of the virtual array is

Definition 1For a sensor position setSsparse array,its difference coarrayDis defined as

Definition 2[34] A sparse arraySdegrees of freedom is the difference coarrayDcardinality.

Definition 3[34] For a sparse arrayS,letUrepresent the difference coarrayDmaximum consecutive segment.TheUcardinality is referred to asSuniform DOFs.

3.2 Mutual coupling

The received signal in (28) does not consider the sensor mutual coupling.However,in practice,the small separation mutual coupling effect cannot be neglected.After considering the mutual coupling effect,(28) can be viewed as

whereCrepresents anN×Nmutual coupling matrix.

TheCexpression is rather complicated35].In ULA,Ccan be approximated by aB-banded symmetric Toeplitz matrix [36] as

Definition 4An arraySweight functionw(l) is defined as the sensor pair number that leads to coarray indexl.

The weight functionw(l) for any linear array withNsensors [35] satisfies

Definition 5For a sensor number,the coupling leakage[35] is defined as the energy ratio:

3.3 Array structure

In this subsection,MISC array based on maximum interelement spacing constraint will be introduced.These arrays have many advantages.First,their sensor locations and the DOFs can be expressed in a closed form.In addition,the MISC arrays difference coarrays are holefree.Most importantly,MISC arrays possess less mutual coupling effect and a higher DOF.

For an arbitrary sensor number,the MISC arrays are constructed by the given interelement spacings.Denote the maximum interelement spacing asD,and the associated interelement spacing set asA.Specifically,MISC arraysDandAare defined as

where ⎿·」 denotes the integral part of the rational number in the square brackets.

The MISC array minimum sensor number is 5,N≥5.Dincreases asNincreases.

The MISC array sensor position set is

The MISC array sensor locations are shown inFig.3.SinceDis decided byN,for an arbitrary sensor,MISC arrays have sensor positions closed-form expressions.

Fig.3 MISC array configuration

3.4 Array analysis

In this subsection,we analyze the performance of MISC array and compare them with some other arrays.It is mainly analyzed from two aspects,which are DOFs and weight functions.The coprime arrays DOFs are calculated instead of uniform DOFs.

(i) Uniform DOFs

For the integer setSMISC,the MISC array difference set is provided by (N−D+3)D+7 and (N−D+3)D−7 consecutive set

The uniform DOFs of the MISC array is

SubstitutingDin (46) with (42) yields

where the approximation does not alter the magnitude orders.

For comparison,the DOFs of arrays,such as ULA,CA,and INA are described below.For ULA,the uniform DOFs are calculated.The ULA uniform DOFs array is

For the coprime array,the zeroth position element can be shared,so there are (N′+N′′−1) elements

The INA provides hole-free difference coarrays [26].This indicates that the MISC array is sparser and has a larger array aperture than the improved nested array,thus it is much less sensitive to mutual coupling effects.The uniform DOFs for the improved nested arrays are given by

From (48),the MISC array uniform DOFs has theN2magnitude order.Therefore,the MISC array has a higher uniform DOFs than the INA.At the same time,the DOFs of MISC array is higher than that of the coprime array,and much higher than that of the ULA.

(ii) Weight function

MISC array’s another advantage is less mutual coupling.For mutual coupling effects,the weight functions at small separations are more important [37].The first three weight functions,w(1),w(2),andw(3),provide a major influence on the mutual coupling,andw(1) has the greatest influence [34].

According to the interelement spacing setAMISCand the weight function definition,its weight functionsw(1),w(2),andw(3) are

In comparison,the first three weight functions for ULA are

and those for the coprime array are

For improved nested arrays,DandAare calculated whenNis small,that is

For INAs,the first three weight functions are

Throughw(1),w(2),andw(3),the mutual coupling effects of the MISC array is compared with two other sparse arrays.Although the coprime array provides a smallerw(2) value,there are holes in the difference coarray.The MISC array is sparser than the improved nested array from the weight functions,so that the mutual coupling is greatly reduced.

4.Quaternion based DOA estimator

4.1 Quaternion spectral matrix

The spectral matrix [38] is proved by second order automoments and cross-moments of all antenna sensors for scalar-sensor arrays.For vector-sensor array,quaternion spectral matrix (QSM),which is the equivalent second order representation,Ω ∈HN×Nis

whereX∈HNis the quaternion observation vector from(28).Replacing (28) in (59) and using (3),Ωcan be expressed as

Assume that the decorrelation between sources themselves and between the noise and the sources,and the quaternion spectral matrix is

where

and ΩN=E{NN◁} is a matrix containing noise second order statistics.In (61),ΩSis the signal part andis themth source power.

The vector sensor arrayX∈HNoutput can be expressed because of the complex-valued outputsX1,X2∈CNof the two components as

The quaternion vector transpose-conjugateXcan be described by complex vectors transpose-conjugatesX1,X2as

where the superscript H represents the conjugate transpose.Introduce (63) and (64) into (59),Ωis

In (65),X1,X2are i-complex vectors and j and i multiplication is not commutative.In (65),for the two vector array components,auto-covariance and cross-covariance matrices mean that QSM contains all the second-order information intrinsically.The QSM noise-free part becomes a Toeplitz matrix [17].Assuming the data (see (28)) noise component is spatially nonpolarized and white,the noise part is a real diagonal matrix and the diagonal entries indicate theNsensors noise power.

4.2 Subspace method

(i) Quaternion vector orthogonality

The received polarization components data of the electromagnetic vector sensor arrays are arranged in order and combined into a long vector data in the complex domain.The advantage of this is that the data can be placed in the complex domain and the complex algebra can be used for operations.However,the polarization components vector structure is destroyed because the data model in the complex domain does not take into account the orthogonality between the polarization components and just stacks the data into a long vector.The data is synthesized into a quaternion and the inherent orthogonality can accurately describe orthogonality between the received components in the quaternion domain instead of simply stacking up the data in the complex domain.

It can be seen that modeling in the quaternion domain can describe the polarization information more completely and accurately than modeling in the complex domain.The quaternion orthogonality has a stronger constraint than the complex one [17]. Theoretically quaternion orthogonality or long vector orthogonality are identical only if sources have identical polarizations or if there is only one source.Otherwise,these two methods obtain different results. Compared to the long vector method,the quaternion vector orthogonality to estimate the signal subspace improves estimation accuracy.The signals energy can be more precisely concentrated on a quaternion vector orthogonal basis than on a long vector one.

(ii) Quaternion eigenvalue decomposition

Eigen-structure methods are based on the vector space decomposition using energy criteria in orthogonal subspaces.Researchers in areas such as quantum mechanics[39],vector-signal processing [40],and color image processing [41] pay attention to singular value decomposition and eigenvalue decomposition (EVD) of quaternion matrices.In this paper,the quaternion eigenvalues and eigenvectors are utilized for right eigenvalues and eigenvectors.The quaternion spectral matrixestimation is proved by statistical average

whereTis the total of snapshots,Xsamis receiving array data at each snapshot andsamis the snapshot index.QSM becomes quaternion Hermitianby construction.Quaternion Hermitian matrix eigenvalues are realvalued [42].Quaternion spectral matrix becomes

whereλnrepresents the real eigenvalue andunrepresents theNorthonormal quaternion eigenvector.

By recognition of (61) with (67),the firstMeigenvalues correspond to the signal part and the restN−Mcorrespond to the noise part.AndMis a priori known.The QSM can be expressed as

withA∈HN×Mincluding theMsource vectors and Σ ∈RM×Ma diagonal matrix including the sources powers.Assuming that noise is not polarized,its covariance matrix is diagonal and real:ΩN=2σ2IN,where 2σ2is the noise power.TheΩEVD is

withU=[u1,···,uN]∈HN×Nincluding the quaternionvaluedNeigenvectors ofΩandD∈RN×N,the eigenvalues diagonal matrix.

(iii) Quaternion MUSICUS∈HN×MUG∈HN×(N−M)

Define two matrices and as

USincludes the signal subspace eigenvectors andUGincludes the noise subspace eigenvectors.By multiplying(68) on the right,we get

QSM can also be indicated according toUSandUGas

whereDS=diag{λ1,···,λM}.The noise power is equal for all array sensors,therefore the lastN−Meigenvalues are 2σ2.Replace (73) in (72),and use the vectors orthogonality betweenUSandUG,and the relation is obtained

which implies

If (75) is multiplied on the right by (A◁UG)◁,(75) can be expressed using columns ofAas

for all {ϕm,θm,γm,ηm} sets corresponding toMsignal sources parameters. ΠN=UGUG◁∈HN×Nis the noise subspace projector.In reality,this projector estimation,results from the estimated spectral matrixEVD.

The quaternion DOA estimation based on vector MISC array is then calculated via projecting the quaternion steering vectoram(ϕm,θm,γm,ηm)∈HN,

A quaternion estimation is similar to the scalar MUSIC algorithm.Equation (78) has maxima for {ϕm,θm,γm,ηm} sets corresponding to signal sources.Varying φm,θm,γm,ηm,a 4D hypersurface is calculated. The ϕm,θm,γm,ηmestimation values are the maxima coordinates on this hypersurface.The firstMmaxima correspond to theMsignal sources.The ϕm,θm,γm,ηmestimation process is quasi-unsupervised. In this paper,the DOA estimation of the signal is mainly concerned and the polarization of the signal is known.For the model described above,the elevation angle is 90°,and only the azimuth angle needs to be estimated.

4.3 Computational complexity

For long vectors and quaternion algorithms,this subsection addresses the computational complexity,only focusing on the covariance matrix estimation.As it indicates repetitive operations,this procedure best indicates the two algorithms’ complexity difference.The method complexity is evaluated regarding memory traffic,memory requirements,and basic arithmetical operations:real numbers addition (A),multiplication (M) and division (D).

Consider a two-component vector sensor arrayeyandez. A snapshot is proved by two complex vectorsX1,X2∈CN. The observation quaternion formX∈HNhas the expression proved by (63) and the long vector representation

The corresponding covariance matrices are

Averaging overTthe covariance matrix is used for estimation:

EachΩsammatrices includeN2quaternion entries and can be expressed for 4N2real fields machine memory,while thematrices have complex entries,corresponding to 8N2real values.For the data covariance model,the quaternion algorithm reduces by half the memory requirements,resulting in memory traffic operations diminution byapproximately and a proportional speed gain.

Evaluate the basic arithmetical operations number for covariance matrix estimation.Each quaternion entries ofΩsamis a result of two quaternions multiplication.It implies 16 real (M) and 12 real (A),and the total number of operations of the whole matrix are 16N2(M) and 12N2(A).16N2(M) and 8N2(A) are calculated for the complex matrix.Thus,the summationΩhave a total of 16N2T(M) and 16N2T−4N2(A) whilerequires 16N2T(M) and 16N2T−8N2(A).The division byTmeans 4N2real numbers (D) for quaternions and 8N2(D) for long vectors.

Regarding the real values elementary operations number,the quaternion model needs 4N2(A) and 4N2(D) less than the long vector model.The division computational complexity is several times more important than addition,meaning higher computational cost for long vectors.

The quaternion eigenvectors computation of the estimated matrix Ω ∈HN×Ncan be executed by complex numbers or quaternions.The complex numbers methods diagonalize theΩcomplex adjoint matrix,a 2N×2Ncomplexvalued matrix [43].In [44],working in quaternion domain enhances the algorithms convergence speed compared to the complex method.This heightens the idea that the quaternions can enhance the algorithms’ performance.The quaternion algorithms reduce computational efforts because of compact data handling.

5.Simulation results

In this section,numerical examples imply the superiority of the vector arrays based on quaternion over the scale arrays in terms of weight functions,mutual coupling matrices,and DOA estimation performance.Assume that the source number is known.For evaluating the results,the DOA root mean square error (RMSE) is defined as a 100 independent trials average:

5.1 Weight functions and mutual coupling matrices

In the simulation example,the weight functions and mutual coupling matrices of the MISC array [29],the ULA[22],the CA [25],and the INA [24] are compared.For all these arrays,we consider that the number of sensors is 11.For the coprime array,the parameters areN′=5 andN′′=7.Here,the mutual coupling model (38) is described byc1=0.05eiπ/3,B=100,andcl=c1e−i(l−1)π/8/lfor 2 ≤l≤B.Fig.4indicates the weight functions of four kinds of arrays.

Fig.4 Weight functions

InFig.4,the ULA has the largest weight functions(w(1)=10,w(2)=9,w(3)=8).The improved nested array exhibits larger weight functions (w(1)=5,w(2)=4,w(3)=3),due to the dense inner ULA.The coprime array provides smaller weight functions (w(1)=w(2)=w(3)=2) because of sparser configuration.The weight functions (w(1)=1,w(2)=2,w(3)=2) are even smaller for the MISC array.

Fig.5indicates the mutual coupling matrices magnitudes.The blue color indicates less energy.The nested array suffers from the severest mutual coupling effect.The MISC array and CA suffer from less mutual coupling effect.

Fig.5 Mutual coupling matrices

5.2 DOA estimation in the absence of mutual coupling

In the simulation example,in the absence of mutual coupling,the DOA estimation performance is compared among the MISC array,the ULA,the CA,and the INA.For all arrays,sensors number is 11.The MUSIC search interval is 0.01° and the incident angle is 30°.The phase difference of the polarization signal is 0 and the constant mode ratio is 0.5.The simulations analyze the RMSE performance about the input SNR,and the snapshots number respectively.The fixed parameter setting isT=1024 snapshots.Fig.6displays the DOA estimation RMSE versus the input SNR.

Fig. 6 RMSE of normalized DOA estimates versus the SNR without mutual coupling

All the RMSEs decrease as the SNR increases.When the SNR is higher than −4 dB,RMSEs attain a steady level.Moreover,the RMSE of the MISC array is smaller than that of the other arrays especially based on the quaternion algorithm.

Fig.7displays the DOA estimation RMSE about the snapshots number.All DOA estimations are more stabilized and accurate as the snapshots number increases,and the MISC array is better than other arrays.The overall quaternion-based algorithm is better than similar algorithms no matter how many snapshots are observed.

Fig.7 RMSE of normalized DOA estimates versus the number of snapshots without mutual coupling

5.3 DOA estimation with mutual coupling

In the simulation,in the presence of mutual coupling,DOA estimation performance of the MISC array,the ULA,the CA,and the INA are compared.The sensors number is 11 and the incident angle is 30°.The RMSE performance about the input SNR,the snapshot number,and the coupling coefficientc1are analyzed.The DOA estimation RMSE about the SNR is displayed inFig.8.The quaternion algorithm for vector arrays exhibits better performance than for scale arrays.The three arrays outperform the ULA against mutual coupling effect.

Fig.8 RMSE of normalized DOA estimates versus the SNR with mutual coupling

InFig.9,the DOA estimation RMSE about the snapshots number is displayed.The RMSE is reduced rapidly for the arrays,as the snapshots number increases.For the quaternion algorithm,the RMSE results are reduced rather stably.The RMSE about |c1| is displayed inFig.10.Along with the |c1| increase,the RMSE increases for any array.More severe mutual coupling effect is introduced by a higher value of |c1|.The CA has a better performance when |c1| is larger than 0.1.

Fig.9 RMSE of normalized DOA estimates versus the number of snapshots with mutual coupling

Fig.10 RMSE of normalized DOA estimates versus the c with mutual coupling

5.4 Other aspects of simulation analysis

(i) CRB of the designed method

The CRB of the design method is analyzed.In order to highlight the design approach,it is taken out separately,and the approach is very close to its CRB.SeeFig.11for details.

Fig.11 CRB of the designed method

(ii) Influence of the minimum scan interval

The effect of the minimum scan interval is simulated and analyzed,as shown inFig.12.The simulation parameters are the same as before,using three scanning intervals,which are 0.01°,0.05°,and 0.1° respectively.The incident angle is 30°.The smaller the interval,the more accurate the estimation,but the search time will also be longer.Therefore,keep scan intervals as small as possible,as computer power permits.Since the angle corresponding to the maximum value is directly selected as the estimate value in the search process,the quantization error is about half of the interval.

Fig.12 Influence of minimum scan interval for designed method

(iii) Relationship between the probability of success and the SNR

In this section,the incident angle is 30°.DOA is estimated successfully if RMSEs are less than 0.05.Simulation parameters are the same as before,and a 0.01 interval is used.The simulation results are shown inFig.13.As the SNR increases,the probability of success increases,and the quaternion algorithm has a better performance.

Fig.13 Relationship between the probability of success and the SNR

(iv) Direction finding ability of the designed method for coherent sources

The performance of the algorithm in the case of coherent sources is analyzed by simulation. The incident angles of the two sources are 40° and 60° respectively.These two sources are coherent sources with the same signal form but slightly different polarization modes.As can be seen from the spectrum inFig.14,this method can distinguish sources but traditional methods cannot.It can be seen that the quaternion-based algorithm designed in this paper has a better performance.

Fig.14 Spectrum

6.Conclusions

In this paper,the combination of vector sensor array and quaternion algorithm shows advantages:higher DOFs,mutual coupling effect reduction,signals polarization characteristic maintenance,and computational burden reduction.For a given sensors number,the DOFs and the sensor locations in MISC array are uniquely determined and can be described via closed-form expressions.The quaternion data model based on vector arrays is established,which extends the scalar array into vector array.Based on the model,a quaternion MUSIC algorithm based on vector MISC arrays is proposed for DOA estimation.The algorithm combines the advantages of the quaternion model and the vector array to enhance the DOA estimation performance. The quaternion model has a lower RMSE.Coprime arrays also have good results under mutual coupling conditions.Analytical simulations are operated to certify the capability of the algorithm.Overall,the vector MISC array based on quaternion has a good effect.